THERMODYNAMICS Course No: ME 209 Department: Mechanical - - PowerPoint PPT Presentation

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THERMODYNAMICS Course No: ME 209 Department: Mechanical Engineering Instructor:

• U. N. Gaitonde

Lecture 23: Open Thermodynamic Systems

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Lecture 23: Open Thermodynamic Systems

• Illustrations of open thermodynamic systems
• A specific case for study and derivation
• Generalisation
• Application to typical engineering systems
• Numerical Exercises

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Illustrations

• Turbines, compressors, pumps
• Fans
• Boilers, condensers, heat exchangers
• Ducts
• Rooms and buildings
• Car
• Human being
• . . . .

An open system is also known as a control volume (CV).

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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A schematic open system

The inflows and outflows could be through ducts or through ports. The flows could also be continuously distributed along the boundary.

˙ Q ˙ Q ˙ WS ˙ WS

• V

˙ mi1 ˙ mi2 ˙ me1 ˙ me2 ˙ me3

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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A schematic open system (contd)

i e

˙ Q ˙ WS

CV It has 1 inlet and 1 outlet.

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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The Situation

The control volume state: volume V (t), Mass M(t), Energy E(t), Entropy S(t) etc.. The fluids at inlet (i) and exit (e) are in local equilibrium. The situation at inlet and exit is 1-dimensional (1D), with everything uniform across the cross-section. Inlet state: area Ai, density ρi, volume vi, energy ei, velocity

Vi, etc.; Vi normal to Ai.

Exit state: area Ae, density ρe, volume ve, energy ee, velocity

Ve, etc.; Ve normal to Ae.

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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The Situation (contd)

The rate of heat transfer to the CV from its surroundings is ˙

Q(t).

The rate at which work is done by the CV is ˙

WS(t). ˙ WS includes all components of work,

except that required for making the fluid flow into and out of the CV.

˙ WS may include, e.g. expansion work, stirrer work, electrical

work, etc..

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Inlet and exit ‘plugs’

i e

˙ Q ˙ WS

CV

t t t+∆t t+∆t

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Inlet and exit ‘plugs’

i

˙ Q ˙ WS

CV a b b’ c c’ d e e e’ f f’

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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A closed system

• Occupies the space [abcdefa] at time t.
• Occupies the space [ab’c’de’f’a] at time t + ∆t.
• No mass flows across the boundaries of this system during

this period.

• So this is a closed system.
• We apply conservation of mass to this system. Then, the first

law, and finally, the second law.

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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The system – intial state

i

˙ Q ˙ WS

CV a b b’ c c’ d e e e’ f f’ The system at time t

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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The system – final state

i

˙ Q ˙ WS

CV a b b’ c c’ d e e e’ f f’ The system at time t + ∆t

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Process and Interactions

i

˙ Q∆t ˙ WS∆t

a b b’ c c’ d e e e’ f f’

M(t)→M(t+∆T ) E(t)→E(t+∆T ) V (t)→V (t+∆T ) S(t)→S(t+∆T ) We Wi

Process and interactions from t to t + ∆t

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Conservation of Mass Msystem(t + ∆t) = Msystem(t) Msystem(t) = MCV(t) + M[bcc’b’] Msystem(t + ∆t) = MCV(t + ∆t) + M[eff’e’] M[bcc’b’] = ρiAiVi∆t M[eff’e’] = ρeAeVe∆t ∴ MCV(t + ∆t) + ρeAeVe∆t = MCV(t) + ρiAiVi∆t MCV(t + ∆t) − MCV(t) ∆t = ρiAiVi − ρeAeVe

So, in the limit as ∆t → 0,

dMCV dt = ρiAiVi − ρeAeVe

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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Conservation of Mass (contd)

We use the nomenclature: Rate of inflow of mass = ˙

mi = ρiAiVi

Rate of outflow of mass = ˙

me = ρeAeVe

So we have:

dMCV dt = ˙ mi − ˙ me

which is the basic form of conservation of mass.

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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First law - for the system ∆E = Q − W

But

∆E = Esystem(t + ∆t) − Esystem(t) Esystem(t) = ECV(t) + E[bcc’b’] Esystem(t + ∆t) = ECV(t + ∆t) + E[eff’e’] E[bcc’b’] = (ρiAiVi∆t)ei E[eff’e’] = (ρeAeVe∆t)ee ∴ ∆E = ECV(t + ∆t) + (ρeAeVe∆t)ee − ECV(t) − (ρiAiVi∆t)ei ∆E = ECV(t + ∆t) − ECV(t) + ˙ meee∆t − ˙ miei∆t

We have: Q = ˙

Q∆t

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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First law - for the system (contd) W = ˙ WS∆t + We + Wi We = peAeVe∆t = peve ˙ me∆t Wi = −piAiVi∆t = −pivi ˙ mi∆t ∴ We + Wi = ˙ me(peve)∆t − ˙ mi(pivi)∆t ∴ the first law becomes ECV(t + ∆t) − ECV(t) + ˙ meee∆t − ˙ miei∆t = ˙ Q∆t − ˙ WS∆t − ˙ me(peve)∆t + ˙ mi(pivi)∆t

Transposing and combining terms:

ECV(t + ∆t) − ECV(t) = ˙ Q∆t − ˙ WS∆t + ˙ mi(ei + pivi)∆t − ˙ me(ee + peve)∆t

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde

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First law - for the control volume dECV dt = ˙ Q − ˙ WS + ˙ mi(ei + pivi) − ˙ me(ee + peve)

We now expand

ei + pivi = ui + V 2

i

2 + gzi + pivi = hi + V 2

i

2 + gzi

ee + peve = ue + V 2

e

2 + gze + pive = he + V 2

e

2 + gze

Thus

dECV dt = ˙ Q − ˙ WS+ ˙ mi(hi+ V 2

i

2 +gzi)− ˙ me(he+ V 2

e

2 +gze)

This is a reasonably general form of the first law for open systems.

ME 209 THERMODYNAMICS Lecture 23

• U. N. Gaitonde